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In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided.

Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a selfsimilar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009) [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.

In this article, a classification of continuous, linearly intertwining, symmetric $L_p$-Blaschke ($p>1$) valuations is established as an extension of Haberl's work on Blaschke valuations. More precisely, we show that for dimensions $n \geq 3$, the only continuous, linearly intertwining, normalized symmetric $L_p$-Blaschke valuation is the normalized $L_p$-curvature image operator, while for dimension $n = 2 $, a rotated normalized $L_p$-curvature image operator is an only additional one. One of the advantages of our approach is that we deal with normalized symmetric $L_p$-Blaschke valuations, which makes it possible to handle the case $p=n$. The cases where $p \neq =n$ are also discussed by studying the relations between symmetric $L_p$-Blaschke valuations and normalized ones.